A central bank sets interest rates on a guess about next year's inflation. A warehouse stocks shelves on a guess about next week's demand. A grid operator fires up a power station on a guess about this afternoon's load. Every one of these is a forecast: a statement about the future values x_{T+1}, x_{T+2}, \dots of a series, made from everything observed up to the present time T. This page answers the sharpest possible version of the question: given an ARMA model and the data so far, what single number should I predict, and why that one?

The best forecast is the conditional expectation

Fix a target: we want the forecast \hat x_{T+h} that minimises the mean squared error \mathbb E\big[(x_{T+h} - \hat x_{T+h})^2\big]. There is a clean, universal answer, and it does not depend on the model being ARMA at all:

So "forecasting" reduces to "compute a conditional expectation". For an ARMA model that computation turns into a tidy recipe you can run by hand.

The recipe: plug in the past, zero out the future

Take the model equation, write it for time T+h, and apply the conditional expectation term by term. Three simple rules do all the work:

• Known past values x_T, x_{T-1}, \dots — leave them as they are (we know them).
• Future values x_{T+1}, \dots, x_{T+h-1} — replace each with its own forecast (recurse).
• Future shocks \varepsilon_{T+1}, \dots — replace with 0, their mean, since they are unpredictable; past shocks become the model residuals we already estimated.

Applying this walking forward one step at a time — h = 1, then feeding that into h = 2, and so on — generates the whole forecast function \hat x_{T+h} recursively. It is the same substitution every time; only the horizon grows.

Worked example — forecasting an AR(1)

Take the mean-zero model x_t = \phi\, x_{t-1} + \varepsilon_t and stand at time T knowing x_T. Apply the recipe:

• One step: \hat x_{T+1} = \mathbb E[\phi x_T + \varepsilon_{T+1}\mid \text{past}] = \phi\, x_T (the future shock zeroes out).
• Two steps: \hat x_{T+2} = \phi\, \hat x_{T+1} = \phi^2 x_T (recurse on the one-step forecast).
• h steps: the pattern is immediate —

With \phi = 0.6 and x_T = 10, the forecasts run 6,\ 3.6,\ 2.16,\ \dots — each a fixed fraction of the last, gliding back toward zero. If the model carries a mean \mu, forecast the deviation and add it back: \hat x_{T+h} = \mu + \phi^h (x_T - \mu).

Forecasts decay to the mean

Because |\phi| < 1 for a stationary series, \phi^h \to 0 and the forecast function slides smoothly to the unconditional mean as the horizon grows: \hat x_{T+h} \to \mu. This is a general fact for any stationary ARMA — the memory of the last observed value fades, and far enough out the best guess is simply "the long-run average". The chart shows an observed series (solid) and its forecast path (the smooth curve) flattening to the mean line: the further ahead you look, the less the present tells you, and the forecast admits it by reverting to the mean.

The rate of the flattening is the persistence: a near-unit-root series (\phi \approx 0.95) coasts back slowly over many steps, while a weakly dependent one (\phi \approx 0.3) snaps to the mean almost at once. An MA(q) model is even blunter — its forecast is exactly the mean for every horizon beyond q, because the model has no memory past q lags.